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Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of a compact Lie group" (Pub. IHES, 1968), where a decomposition of $\operatorname{Spec}R(G)$ into connected components is obtained. Here I mean complexified representation ring. If $\pi_0(G)=\mathbb{Z}/2\mathbb{Z}$, one of the connected components is always of the form $\operatorname{Spec}(R(S)^{W})$, where $S$ is a maximal torus of $G^0$, $W_S=N_G(S)/S$ is the Weyl group. (Note the normalizer is taken in the entire group.)

Frequently, it turns out that $R(S)^{W}$ is the representation ring of some other, connected, group. For example, if $G=\operatorname{O}_{2n}$ and $S$ is a maximal torus of $\operatorname{SO}(2n)$, then $R(S)^{W}$ is a type $B_n/C_n$ representation ring; conjugation by $C_\epsilon$, where $\epsilon$ is the permutation matrix for $(1, 2n)$, affords an element of the Weyl group changing a single sign. For example, when $n=1$, we get $R(S)^W=R(S)^{C_\epsilon}=\mathbb{C}[z+z^{-1}]$.

What is a reference, or at least a name, for this phenomenon in general? This is not exactly folding of the Dynkin diagram, because the rank is unchanged.

Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of a compact Lie group" (Pub. IHES, 1968), where a decomposition of $\operatorname{Spec}R(G)$ into connected components is obtained. Here I mean complexified representation ring. If $\pi_0(G)=\mathbb{Z}/2\mathbb{Z}$, one of the connected components is always of the form $\operatorname{Spec}(R(S)^{W})$, where $S$ is a maximal torus of $G^0$, $W_S=N_G(S)/S$ is the Weyl group. (Note the normalizer is taken in the entire group.)

Frequently, it turns out that $R(S)^{W}$ is the representation ring of some other, connected, group. For example, if $G=\operatorname{O}_{2n}$ and $S$ is a maximal torus of $\operatorname{SO}(2n)$, then $R(S)^{W}$ is a type $B_n/C_n$ representation ring; conjugation by $C_\epsilon$, where $\epsilon$ is the permutation matrix for $(1, 2n)$, affords an element of the Weyl group changing a single sign. For example, when $n=1$, we get $R(S)^W=R(S)^{C_\epsilon}=\mathbb{C}[z+z^{-1}]$.

What is a reference, or at least a name, for this phenomenon in general? This is not exactly folding of the Dynkin diagram, because the rank is unchanged. (Rather, it is the other connected component that is the representation ring of a group whose Dynkin diagram is obtained by folding.)

Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of a compact Lie group" (Pub. IHES, 1968), where a decomposition of $\operatorname{Spec}R(G)$ into connected components is obtained. Here I mean complexified representation ring. If $\pi_0(G)=\mathbb{Z}/2\mathbb{Z}$, one of the connected components is always of the form $\operatorname{Spec}(R(S)^{W})$, where $S$ is a maximal torus of $G^0$, $W_S=N_G(S)/S$ is the Weyl group. (Note the normalizer is taken in the entire group.)

Frequently, it turns out that $R(S)^{W}$ is the representation ring of some other, connected, group. For example, if $G=\operatorname{O}_{2n}$ and $S$ is a maximal torus of $\operatorname{SO}(2n)$, then $R(S)^{W}$ is a type $B_n/C_n$ representation ring; conjugation by $C_\epsilon$, where $\epsilon$ is the permutation matrix for $(1, 2n)$, affords an element of the Weyl group changing a single sign. For example, when $n=1$, we get $R(S)^W=R(S)^{C_\epsilon}=\mathbb{C}[z+z^{-1}]$.

What is a reference, or at least a name, for this phenomenon in general? This is not exactly folding of the Dynkin diagram, because the rank is unchanged.

Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of a compact Lie group" (Pub. IHES, 1968), where a decomposition of $\operatorname{Spec}R(G)$ into connected components is obtained. Here I mean complexified representation ring. If $\pi_0(G)=\mathbb{Z}/2\mathbb{Z}$, one of the connected components is always of the form $\operatorname{Spec}(R(S)^{W})$, where $S$ is a maximal torus of $G^0$, $W_S=N_G(S)/S$ is the Weyl group. (Note the normalizer is taken in the entire group.)

Frequently, it turns out that $R(S)^{W}$ is the representation ring of some other, connected, group. For example, if $G=\operatorname{O}_{2n}$ and $S$ is a maximal torus of $\operatorname{SO}(2n)$, then $R(S)^{W}$ is a type $B_n/C_n$ representation ring; conjugation by $C_\epsilon$, where $\epsilon$ is the permutation matrix for $(1, 2n)$, affords an element of the Weyl group changing a single sign. For example, when $n=1$, we get $R(S)^W=R(S)^{C_\epsilon}=\mathbb{C}[z+z^{-1}]$.

What is a reference, or at least a name, for this phenomenon in general? This is not exactly folding of the Dynkin diagram, because the rank is unchanged.

Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in this paper of his, where a decomposition of $\operatorname{Spec}R(G)$ into connected components is obtained. Here I mean complexified representation ring. If $\pi_0(G)=\mathbb{Z}/2\mathbb{Z}$, one of the connected components is always of the form $\operatorname{Spec}(R(S)^{W})$, where $S$ is a maximal torus of $G^0$, $W_S=N_G(S)/S$ is the Weyl group. (Note the normalizer is taken in the entire group.)

Frequently, it turns out that $R(S)^{W}$ is the representation ring of some other, connected, group. For example, if $G=\operatorname{O}_{2n}$ and $S$ is a maximal torus of $\operatorname{SO}(2n)$, then $R(S)^{W}$ is a type $B_n/C_n$ representation ring; conjugation by $C_\epsilon$, where $\epsilon$ is the permutation matrix for $(1, 2n)$, affords an element of the Weyl group changing a single sign. For example, when $n=1$, we get $R(S)^W=R(S)^{C_\epsilon}=\mathbb{C}[z+z^{-1}]$.

What is a reference, or at least a name, for this phenomenon in general? This is not exactly folding of the Dynkin diagram, because the rank is unchanged.

Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of a compact Lie group" (Pub. IHES, 1968), where a decomposition of $\operatorname{Spec}R(G)$ into connected components is obtained. Here I mean complexified representation ring. If $\pi_0(G)=\mathbb{Z}/2\mathbb{Z}$, one of the connected components is always of the form $\operatorname{Spec}(R(S)^{W})$, where $S$ is a maximal torus of $G^0$, $W_S=N_G(S)/S$ is the Weyl group. (Note the normalizer is taken in the entire group.)

Frequently, it turns out that $R(S)^{W}$ is the representation ring of some other, connected, group. For example, if $G=\operatorname{O}_{2n}$ and $S$ is a maximal torus of $\operatorname{SO}(2n)$, then $R(S)^{W}$ is a type $B_n/C_n$ representation ring; conjugation by $C_\epsilon$, where $\epsilon$ is the permutation matrix for $(1, 2n)$, affords an element of the Weyl group changing a single sign. For example, when $n=1$, we get $R(S)^W=R(S)^{C_\epsilon}=\mathbb{C}[z+z^{-1}]$.

What is a reference, or at least a name, for this phenomenon in general? This is not exactly folding of the Dynkin diagram, because the rank is unchanged.